Was Einstein lucky when not considering twin paradox as paradox?

  • #151
stevendaryl
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For good understanding: it was also expressed as the "impossibility to detect absolute motion", because the same laws of physics are observed in systems that are moving relative to each other.

Everyone observes the same laws of physics, no matter what their state of motion. Everyone sees the same universe, after all. It's funny that you say that there is no need to clarify what "valid" means, when it sure seems to me that the concept is muddled.

The issue is in terms of the mathematical form of the equations of motion expressing those laws. For a given law, there may be a set of coordinate systems for which that law takes an exceptionally simple form. If by "valid coordinate system" you mean "a coordinate system in which the laws look simplest", then there might be a limited number of valid coordinate systems. But to me, that's a bizarre criterion. If you do Newton's mechanics using polar coordinates, the form of the equations of motion are changed. There are is an additional term in the equations, sometimes called "centrifugal force":

Instead of [itex]m \frac{d^2 r}{dt^2} = F^r[/itex], you get [itex]m (\frac{d^2 r}{dt^2} - m r (\frac{d\theta}{dt})^2) = F^r[/itex]

Does that mean that polar coordinates are not "valid"?

The twin "paradox" in SR illustrates nicely that non-inertial reference systems are not valid systems for application of the Lorentz transformations with SR.

I think that is not a very clear way to put it. The Lorentz transformation is a coordinate transformation connecting two systems of coordinates. If [itex](x,t)[/itex] is an inertial coordinate system, and [itex](x',t')[/itex] is a noninertial coordinate system, then they are not related by a Lorentz transformation, in the same way that rectangular coordinates are not related to polar coordinates through a Galilean transformation. But that doesn't say anything about the "validity" of noninertial coordinates.

The Lorentz transformations are mathematics. The physical content comes in when you operationally define two coordinate systems (for example, you define how they would be set up using standard clocks and measuring rods in various states of motion). Then the claim that two operationally defined coordinate systems are related by a Lorentz transformation is an empirical claim.

However, according to 1916 GR, reference systems in any state of motion must be valid in that sense, with the physics of that theory - as Einstein defended in 1918:
"It is certainly correct that from the point of view of the general theory of relativity we can just as well use coordinate system K' as coordinate system K."
I discussed that issue in post #140 in this thread.

Yes, that's the quote that I'm saying is very much misleading. Yes, GR allows you to use coordinate system K', but so does SR. Using system K' in SR is no more problematic than using polar coordinates in Newton's mechanics.
 
  • #152
Dale
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This is just going around in circles. Time to move on.
 

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